PUC Science 2nd PUC Class 12 - Karnataka Board PUC Important Questions for Mathematics

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Find `intsqrtx/sqrt(a^3-x^3)dx`

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

Find the projection of the vector `hati+3hatj+7hatk`  on the vector `2hati-3hatj+6hatk`

The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A

Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.

If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`

Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk`  are coplanar.

If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `